Finiteness of certain local cohomology modules

Cofiniteness of the generalized local cohomology modules $H^{i}_{mathfrak{a}}(M,N)$ of two $R$-modules $M$ and $N$ with respect to an ideal $mathfrak{a}$ is studied for some $i^{,}s$ witha specified property. Furthermore, Artinianness of $H^{j}_{mathfrak{b}_{0}}(H_{mathfrak{a}}^{i}(M,N))$ is investigated by using the above result, in certain graded situations, where $mathfrak{b}_{0}$ is an ideal of $R_{0}$ and $mathfrak{a}=mathfrak{a}_{0}+R_{+}$ such that $mathfrak{b}_{0}+mathfrak{a}_{0}$ is an $mathfrak{m}_{0}$-primary ideal.